Kosevich A.M. The crystal lattice

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334

14 Dislocation Dynamics

Fig. 14.3 Two types of dislocation motion in the ﬁeld of the Peierls po-

tential: (

) the dislocation vibrates in one potential valley; (

) the disloca-

tion forms a kink moving along the x-axis.

It is clear that (14.4.6) is an inhomogeneous sine-Gordon equation, so that its soli-

tons are associated with the dislocation kink s.

Unfortunately, the local equation of m otion for the dislocation element (14.4.6) can-

not be derived consistently from the equation of motion (14.4.1). In a crystal there are

no speciﬁc interactions decreasing so fast in space that it be possible to pass from the

integral equation (14.4.1) to the differential equation in partial derivatives (14.4.6).

The elastic stresses causing the interaction of various parts of the same dislocation

decrease very slowly with distance, so that such dislocation parameters as the mass

m per unit length or the linear tension coefﬁcient T

are not local characteristics of

the dislocation. Although the string model is limited, it is often used to demonstrate

the physical phenomena generated by the dislocation bending vibrations. The model

is attractive primarily because it is simple and enables good results to be obtained.

However, the conclusions made from an analysis of (14.4.6) need to be re-examined.

We shall take the r.h.s. of (14.4.6) as an external force. In the absence of an ex-

ternal ﬁeld (σ = 0), then the dislocation motion has the character of free oscillations

corresponding to the normal modes of string vibrations and having the dispersion law

= ω

+ c

, (14.4.7)

where

= 2πbσ

/(ma), c

= T

/m. (14.4.8)

Let us discuss the dispersion law (14.4.7). In analyzing the waves localized near

the dislocation (Section 12.2) it was noticed that its bending vibrations should have a

propagation velocity that does not actually differ from that of transverse sound vibra-

tions s

. To conﬁrm this conclusion with (14.4.7) we assume c

= s

, which does not

contradict the estimates (14.3.7), (14.3.8). But the dispersion law frequencies (14.4.7)

satisfy the inequality

ω > c

k = s

k. (14.4.9)

The wave running along the dislocation with wave vector k and frequency (14.4.8)

cannot be localized near the dislocation because it inevitably excites bulk vibrations

whose frequencies satisfy the same condition (14.4.9). Thus, the string vibration with

14.4 Equation for Dislocation Motion

335

the dispersion law (14.4.7), if it exists, has the character of a quasi-local vibration. Un-

fortunately, it cannot be pronounced. Indeed, it is easily seen that the connection of the

dislocation lin e vibrations with the vibrations of crystal lattice atoms is characterized

by the same parameter that determines the eigenfrequency ω

We note that the periodic potential in whose valleys the dislocation vibrates is cre-

ated by the crystal atoms. Thus, it is more natural in (14.4.6) to write the periodic term

in a different way, i. e.,

∂

∂t

− c

∂

∂x

a ω

2π

sin



2π

η − u



= 0, (14.4.10)

where u

is the long-wave atomic displacement in the vicinity of the dislocation core.

In studying small vibrations, we have instead of (14.4.10)

∂

∂t

−c

∂

∂x

+ ω

η = ω

, (14.4.11)

where there is no small coupling parameter between the dislocation and sound dis-

placements that permits consideration of free dislocation v ibrations in the valley of a

periodic potential relief.

Thus, examining the eigenvibrations of a dislocation-string leads to the conclusion

that (14.4.6) is not a satisfactory model for describing the motion of a free dislocation.

However, th is model is quite applicable for investigating the dislocation vibrations

under the action of an oscillating external force. We consider the dislocation vibrations

under the conditions when the Peierls forces can be neglected (e. g., at high temper-

atures) and the retarding force of the dislocation that is proportional to its velocity

becomes important. In this case the dislocation motion obeys an equation similar to

(14.4.6), but the term app earing from the Peierls potential can be omitted and the dissi-

pative term B(∂η/∂t),whereB is the mu ltiplier depending on the nature of dissipative

forces, is added. As a result we get

∂

∂t

+ B

∂η

∂t

− T

∂

∂x

= bσ

−iωt

. (14.4.12)

We assume that the dislocation is ﬁxed at the two points ±l/2. This ﬁxing can be

produced by a strong interaction of the dislocation core with a poorly mobile impurity

in a crystal. Then, the solution for the forced vibrations of a dislocation segment is the

function

η =

bσ

−iωt



cos kx

cos

−1



, (14.4.13)

where k

=(mω

+ iBω)/T

Equation (14.4.12) and its solution (14.4.13) are the basis for a dislocation theory

of internal friction worked out in detail by Granato and Lucke (1956).

336

14 Dislocation Dynamics

14.5

Vibrations of a Lattice of Screw Dislocations

Consider a superstructure formed by a lattice of parallel screw dislocations. By dislo-

cation lattice we mean a system o f parallel screw dislocation s oriented along the z-axis

and intersecting the xy plane at discrete periodically arranged points, forming a 2D

lattice, the unit cell of which has area S

: S = NS

,whereS is the cross-sectional area

of the sample in the xy plane and N is the number of dislocations. The coordinates of

these points in the equilibrium lattice are

x(n)=R

∑

, n =(n

, n

,0), (14.5.1)

where d

(α = 1, 2) are the basic translation vectors of the lattice (d

∼ d is the

distance between neighboring dislocations: S

= d

We intend to use a simple one-component scalar model of vibrations in which it is

assumed that all the atoms are displaced only in one direction. The basis of using such

a model is the fact that a static screw dislocation in the isotropic media produces the

scalar ﬁeld of displacements w along the z-axis. The model gives a correct description

of the elastic ﬁeld created in an isotropic medium by parallel screw dislocations. The

solution of such a problem in a real vector displacement scheme can in principle be

found analytically, but it permits obtaining the dispersion relation of the dislocation

lattice only in implicit form. For the sake of simplicity we restrict ourselves to th e

scalar model.

If there are rectilinear screw dislocations directed along the z axisx, then the elastic

ﬁeld is more conveniently described not by the displacement w along the z-axis but

by the distortion and velocity of the displacements as functions of the coordinate and

time. Following (13.1.10) and (13.1.12), for describing the shear ﬁeld of screw dislo-

cations we introduce a distortion vector h and a stress vector σ = Gh (G is the shear

modulus):

h = grad w, h

= ∇

w =

∂w

∂x

, i = 1, 2, 3, (14.5.2)

and a velocity v: v = ∂w/∂t. Equation (13.1.11) conserves its form

curl h = −α, (14.5.3)

and the density of dislocations α is equal to

α = τb

∑

δ(x − R

where b is the modulus of the Burgers vector and τ is the tangent vector to the dislo-

cation; for a static dislocation it is conveniently chosen as τ(0, 0, −1).

The wave equation for the elastic ﬁeld in the medium b etween dislocations takes

the usual form (14.3.1):

div h −

∂v

∂t

= 0. (14.5.4)

14.5 Vibrations of a Lattice of Screw Dislocations

337

If the dislocations move (vibrate), then (14.5.3), (14.5.4) do not change, but a new

variable of the dislocation structure appears: u =(u

, u

,0) (of course, u = u(n, t)

for the n-th dislocation), which determines the instantaneous coordinate of an element

of the dislocation:

= R(n) + u(n, t) .

The time dependence of the displacement vector u gives the velocity V of an element

of the dislocation (V

= ∂u/∂t) th a t generates a dislocation ﬂu x. The dislocation ﬂux

density vector j arises in the dynamical equation (14.1.3):

∂h

∂t

= grad v + j . (14.5.5)

In the case under consideration the ﬂux density is given by the formula following

from (14.1.4):

= bε

αβ

∑

(n)δ (x − R(n)) , α = 1, 2 , (14.5.6)

where the matrix ε

αβ





−10





. (14.5.7)

Collecting together (14.5.3)–(14.5.5) we obtain the total set of equations describing

the e lastic ﬁeld in the sample if the distribution of dislocations and their ﬂuxes are

known. To close th is system it is necessary to write equations of motion for the dislo-

cations under the inﬂuence of the elastic ﬁelds. The simplest form of such an equation

can be obtained using (14.4.1) and (14.4.4) for rectilinear dislocations:

∂V

∂t

= f

+ S

, α = 1, 2 , (14.5.8)

here m is the effective mass of a unit length of the dislocation (with the order of

magnitude (14.3.7), where R is the distance between dislocations in our case), and f

is equal to

= bε

αβ

= bG ε

αβ

. (14.5.9)

In the case of a curved dislocation line expression (14.5.9) includes the self-force from

different elements of the same dislocation, which is proportional to the curvature of

the dislocation line at the given point. In the analysis of small vibrations the curvature

of the dislocations can not be taken into account, and the force (14.5.9) includes only

the stresses created by the other dislocations.

Usually S is th e force due to the discreteness of the lattice, including dissipative

forces. As we are interested in the dispersion relation for small vibration s, we neglect

the latter and take the force in the form equivalent to that on (14.4.11), namely,

S = −mω

u, (14.5.10)

338

14 Dislocation Dynamics

where ω

is the frequen cy of vibrations of the dislocation string in a valley of the

Peierls relief.

Let us investigate the long-wavelength vibrations of the dislocation lattice, assum-

ing the wavelength of the vib rations is much larger than the lattice period d(dk  1).

In this approximation the distribution of the dislocations can be assumed continuous,

characterized by a density n(x, t). In equilibrium n = n

= 1/S

. The dynamics

of the lattice is governed mainly by the average dislocation ﬂux density, which in the

linear approximation has the nonzero components

(x, t)=bn

αβ

(xt), α = 1, 2, (14.5.11)

where V is the average velocity of the dislocations. The velocity V must be deter-

mined by the equation of motion of the dislocations (14.5.8). We write the equation

of motion with the use of (14.5.9) and (14.5.10):

∂V

∂t

+ ω

αβ

. (14.5.12)

We differentiate (14.5.12) with respect to time and use (14.5.5) and (14.5.11). After

elementary calculations we get

∂V

∂t

+(ω

+ ω

αβ

∂v

∂x

, (14.5.13)

where

. (14.5.14)

The frequency ω

is the analog of the plasma frequency.

We now differentiate (14.5.4) with respect to time and again use (14.5.5):



∆ −

∂

∂t



v = −bn

αβ

∇

. (14.5.15)

Here ∆ is the Laplacian operator.

The pair of equations (14.5.13), (14.5.15) describe the collective dynamics of

the dislocation lattice and the elastic ﬁeld . It is easy to show that the equation

for the “longitudin a l” vibrations of the lattice separates. Indeed, for the variable

P = div V = ∇

it follows from (14.5.13) that

∂

∂t

+ ω

P = 0, ω

= ω

+ ω

, (14.5.16)

where ω

plays the role of the frequency of the longitudinal vibrations of the lattice.

The longitudinal component of the average velocity of the dislocations is derivable

from a potential: V

(1)

= ∇

Q, α = 1, 2. Taking that into account, we see that the

“longitudinal” vibrations of the elastic ﬁeld v = v(z) do not depend on the lattice

14.5 Vibrations of a Lattice of Screw Dislocations

339

vibrations. Thus one branch of collective vibrations (we call it the branch of “longitu-

dinal” vibrations) corresponds to independent oscillations of the elastic ﬁled v= v(z, t)

with the dispersion relation ω = sk

and to compression–rarefaction oscillations of

the dislocation lattice P = P(z, t) with the dispersion relation ω = ω

To describe to “transverse” vibrations we introduce the variable

M = bn

(curl V)

= bn

αβ

∇

. (14.5.17)

The equation for this variable follows from (14.5.13):

∂

∂t

+ ω

M = −ω

∂

∂x

. (14.5.18)

The “transverse” collective vibrations are described by (14.5.18) and the following

equation obtained from (14.5.15) for the function v(x, y, t):



∂

∂t

−

∂

∂x



v = M. (14.5.19)

The compatibility conditions for (14.5.18) and (14.5.19) give the dispersion relation

for a wave with wa ve vector k(k

, k

,0):

−(ω

+ s

)ω

− ω

= 0. (14.5.20)

Equation (14.5.20) has two roots for ω

, which correspond to low-frequency and

high-frequency oscillations. Without writing the trivial expressions for these solutions

in quadratures, we note the following:

Low-frequency branch. For sk  ω

the dispersio n relation has the form

ω =





sk. (14.5.21)

The vibrations are characterized by a transverse sound velocity, the value o f which

is less than the sound velocity s in the medium without the dislocations.

High-frequency branch. For sk  ω

the inertial dislocation lattice is not entrained

in the motion, and one observes only vib rations of the elastic ﬁeld with the usual

sound dispersion relation ω = sk. Finally, in the long-wavelength limit (sk  ω

)

we obtain

= ω





+ k

). (14.5.22)

In comparing the graphs of the two branches of the dispersion relation, one must

be particularly careful in rendering the low-frequency branch. The point is that dis-

persion relation (14.5.20) is valid for λ  a (or ak  1). At large k the dispersion

relation of the lattice manifests a periodic dependence on the quasi-wave vector with

the r eciprocal lattice period G: ω(k )=ω(k + G) . Therefore the dispersion rela-

tion obtained is actually valid in all small neighborhoods of any 2D r eciprocal lattice

340

14 Dislocation Dynamics

Fig. 14.4 Diagram of the dispersion relation : (1) ω = sk, (2) plot of

(14.5.21), (3) expected form of the graph in the short-wavelength region.

vector g,i.e.,fora |k − g|1. Consequently, we are justiﬁed in drawing only the

part of the graphs shown by the heavy solid lines 1 and 2 in Fig. 14.4 for a certain

“good” direction in the reciprocal lattice. The continuation of the graph of the lower

branch at k ∼ π/a and also the indicated crossing of the graphs of the upper branch

at k =(p + 1/2)π/a, p = 1, 2, 3, . . . can be described only on the basis of a study of

the dynamics of the discrete dislocation lattice. That is a subject for a separate study.

We can only state that the graph of the lower branch is closed by the curves illustrated

schematically by the dotted lines 3 in Fig. 14.4. Whether or not there is a band of

forbidden frequencies between the upper and lower branches (gap in the spectrum)

one cannot say on the basis of a long-wavelength treatment. However, one can say

that th e frequency spectrum has a limiting frequency ω

that marks the edge of the up-

per branch of vibrations, which can certainly be manifested in the acoustic resonance

properties of a crystal with a dislocation lattice.

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The Crystal Lattice: Phonons, Solitons, Dislocations, Superlattices, Second Edition. Arnold M. Kosevich

 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

ISBN: 3-527-40508-9

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Index

acoustic vibrations 24

adiabatic approximation 60

analytical critical points 135

anharmonic approximation 285

annihilation operator 171

atom displacement 20

atomic localization 176

bending vibration 31

bending waves 33

Born–Karman conditions 71

boundary between blocks 245

Bragg reﬂection law 12

branches of vibrations 37

Bravais lattice 5

Brillouin zone 8

broken spontaneously symmetry 74

Burgers vector 234

catastrophe 257

classical diffusion 222

climb of a dislocation 241

collecti ve excitations 74

conical point 127

creation operator 171

crossover situation 101

crowdion 40

crystal ground state 74

crystal lattice defect 215

crystal melting 317

damping time 273

de Bour parameter 205

Debye frequency 68

Debye model 68

Debye–Waller factor 197

decaying dispersion law 189

defecton 219

density of states 130

de viator tensor 79

diagram technique 264

dilatation center 230

dipole lattice 101

disclination 241

dislocated crystal 314

dislocation 233

dislocation axis 233

dislocation core 233

dislocation density 298

dislocation dipole 318

dislocation ﬂux density 322

dislocation lattice 336

dislocation loop 235

dislocation pile-up 307

dislocation wall 244

dispersion law 17

dispersion relation 19

displacement vector 60

distortion tensor 78

dumb-bell conﬁguration 217

dynamical matrix 60

edge dislocation 233

effective Hamiltonian 187

effective mass of the dislocation 330

elastic dipole 218

elastic moduli tensor 76

elastic superlattice 153

elementary excitations 171

elliptical point 256

equation of motion 18

equation of phonon motion 198

extended defects 279

forbidden frequencies 153

forbidden gap 153

Frank v ector 242

Frenkel pair 216

Frenkel–Kontorova model 39

frequency spectrum 20

The Crystal Lattice: Phonons, Solitons, Dislocations, Superlattices, Second Edition. Arnold M. Kosevich

 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

ISBN: 3-527-40508-9